Question

Suppose the income elasticity of demand for food is 0.5 and the price elasticity of demand is \(-1.0 .\) Suppose also that Felicia spends \(\$ 10,000\) a year on food, the price of food is \(\$ 2,\) and that her income is \(\$ 25,000\) a. If a sales tax on food caused the price of food to increase to \(\$ 2.50,\) what would happen to her consumption of food? (Hint: Because a large price change is involved, you should assume that the price elasticity measures an arc elasticity, rather than a point elasticity.) b. Suppose that Felicia gets a tax rebate of \(\$ 2500\) to ease the effect of the sales tax. What would her consumption of food be now? c. Ts she better or worse off when given a rebate equal to the sales tax payments? Draw a graph and explain.

Short answer

a. Felicia's food consumption will decrease by approximately 22.22% due to the price increase. b. Her consumption will increase by approximately 4.76% due to the tax rebate. c. Taking both changes into account, her revised food consumption will decrease by approximately 17.46%, indicating she's worse off even with the rebate.

Step-by-step solution

Calculate the new consumption based on price elasticity of demand

First, we apply the formula of arc price elasticity of demand: \( \text{PED} = \frac{\text{\% change in quantity demanded}}{\text{\% change in price}} \). In this case, the \text{PED} value is given as \(-1.0\). We know that the old price was \$2 and the new price is \$2.50. So, the percentage change in price = \( \frac{\text{(new price - old price)}}{\text{(Average of old and new price)}} \times 100\% = \frac{\$0.50}{\$2.25} \times 100\% \approx 22.22\%\). Then, to find the change in quantity, we rearrange the PED formula and plug in the numbers: \text{\% change in quantity demanded} = \( \text{PED} \times \text{\% change in price} = -1.0 \times 22.22\%\approx -22.22\% \). That means her food consumption will decrease by approximately 22.22%.

Calculate the new consumption based on income increment

Felicia gets a tax rebate of \$2500, which increases her income. Using the income elasticity of demand (IED), we can calculate the effect of this income increase on her food consumption. The formula for the IED is: \(\text{IED} = \frac{\text{\% change in quantity demanded}}{\text{\% change in income}} \). The IED for food is given as 0.5. The percentage change in her income is: \( \frac{\text{(new income - old income)}}{\text{(average of old and new income)}} \times 100\% = \frac{\$2500}{\$26250} \times 100\% \approx 9.52\% \). Rearrange the IED formula and plug in the numbers: \(\text{\% change in quantity demanded} = \text{IED} \times \text{\% change in income} = 0.5 \times 9.52\%\approx 4.76\% \). This indicates her consumption will increase by approximately 4.76% due to the income increment.

Deduce if Felicia is better or worse off

From above, Felicia's food consumption decreases by 22.22% due to price increase and it increases by 4.76% due to income increase. The net effect is that her food consumption will change by -22.22% + 4.76% = -17.46%. That means her food consumption will still decrease by ~17.46%, even considering both the price increase and the income increment. So, she is worse off when given a rebate equal to the sales tax payments. You can draw a simple graph with quantity of food on the x-axis, price on the y-axis, and show that after the sales tax and rebate, Felicia's demand curve for food shifts to the left.

Key concepts

Income Elasticity

Income elasticity of demand can be a bit of a puzzle. However, it's really about understanding how consumers adjust their buying habits when their income changes. In Felicia's case, a tax rebate increases her income. We use the Income Elasticity of Demand, which in this case is 0.5, to decide what happens to how much food she buys.

To find out how an income change affects demand, we look at the percentage change in her income and multiply it by the income elasticity.

• When Felicia gets a tax rebate of $2500, her income goes up by 9.52%.
• The income elasticity is 0.5, meaning her food consumption increases, but at a slower rate than her income growth.
• Her food consumption will increase by 4.76% due to her higher income.

This tells us a little about consumer behavior as well—when people earn more, they usually buy a bit more, though not always a lot more, of essentials like food.

Price Elasticity

Price elasticity of demand helps us understand how much the quantity demanded changes when there's a price change. With a price elasticity of -1.0, Felicia's demand for food is quite elastic, which means she is sensitive to changes in food prices.

When a sales tax increases the price of food from $2 to $2.50, it's a significant increase, resulting in a couple of changes:

• The percentage price change is approximately 22.22%.
• This causes Felicia's food consumption to drop by roughly the same rate, 22.22%, because the price elasticity is -1.0.

Price elasticity is important in understanding consumer behavior because it shows us that when prices rise, especially sharply, consumers like Felicia reduce their demand considerably. This is why businesses pay close attention to price elasticity when setting prices.

Consumer Behavior

Consumer behavior delves into the choices people make when circumstances change, like shifts in income or prices. In this scenario, we can see two main factors influencing Felicia's buying decisions: the increase in food prices and her tax rebate.

Despite having more money thanks to the rebate, the bigger picture becomes clear when both income and price influences are combined:

• The price increase leads to a 22.22% drop in consumption.
• The income boost counteracts this by increasing consumption by 4.76%.
• Nevertheless, the net effect is a decrease of 17.46% in her food consumption.

The result? Felicia is worse off even with the rebate. This tells us a significant story about consumer behavior: price increases, especially for essential goods like food, tend to have a stronger impact on consumption than equivalent increases in income. Understanding this can help consumers make better choices, and policymakers design more effective economic interventions.